application of fourth-order cumulants to delay time …...doa estimation of multiple delayed waves...
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Application of Fourth-Order Cumulants to Delay Time and
DOA Estimation of Multiple Delayed Waves by the MMP
Method
Yuichi Shimamura*
Department of Electronic Engineering, Graduate School of Science and Engineering,
Saitama University, Urawa, Japan 338�8570
Kiyomichi Araki
Department of Computer Engineering, Tokyo Institute of Technology, Tokyo, Japan 152-8552
Jun-ichi Takada
International Cooperative Center for Science and Technology, Tokyo Institute of Technology, Tokyo, Japan 152-8552
SUMMARY
This paper proposes an estimation method for the
delay time, the direction of arrival, and the signal strength
of multiple delayed waves, which are important parameters
characterizing the multiple delay characteristics of mobile
radio communication systems. The proposed method is an
estimation method in which the superresolution and higher-
order statistics are used as the guiding principle. The fre-
quency characteristics of the multiple propagation path
environment are observed by a linear array antenna, and
estimated by the modified matrix pencil (MMP) technique,
by generating the virtual correlation matrix from the result
of processing of the fourth-order cumulants. The method
can estimate at least twice as many delayed waves as when
the second-order moment is used, if the number of obser-
vation points on the frequency axis and the number of
antenna elements are the same. A method is also proposed
in which the error in the generation of the virtual correlation
matrix is reduced. The effectiveness of the proposed
method is demonstrated by computer simulation. © 1999
Scripta Technica, Electron Comm Jpn Pt 1, 82(12): 30�39,
1999
Key words: Multiple delayed waves; estimation of
delay time; estimation of direction of arrival; superresolu-
tion method; higher-order statistics.
1. Introduction
At present, mobile communication services are
widely utilized as means of personal communications, and
the demands are still increasing rapidly. As another aspect,
the function of transmitting multimedia information is con-
sidered necessary in nonverbal services, for which increas-
ing demand is expected. From such a viewpoint, in the next
CCC8756-6621/99/120030-10
© 1999 Scripta Technica
Electronics and Communications in Japan, Part 1, Vol. 82, No. 12, 1999Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J81-B-II, No. 4, April 1998, pp. 289�296
* Presently with NTT.
30
generation mobile radio communication systems must
drastically improve their spectral efficiency and transmis-
sion rate by techniques such as separation of the paths from
multiple mobile stations using the same frequency at the
same time [1], and elimination of intersymbol interference.
In order to meet these requirements, adequate identification
or estimation of the multiple propagation paths between the
mobile station and the base station, that is, estimation of the
parameters characterizing the multiple delay charac-
teristics, is crucial.
In order to realize such an estimation procedure, the
superresolution technique and higher-order statistics are
utilized as the guiding principle in the method proposed in
this paper. Typical methods in the superresolution tech-
nique are MUSIC [2] and ESPRIT [3]. The method achieves
higher resolution than traditional methods such as the FFT.
As a delay estimation method belonging to the superreso-
lution technique, the MMP (modified matrix pencil) method
has been presented [4]. The MMP method is a superresolution
method that can realize high estimation accuracy without
using the peak search process as in MUSIC.
Furthermore, only the second-order statistics such as
the mean and variance are usually utilized in traditional
signal processing. The application of higher-order statistics
to the signal processing technique is also being considered
at present [5, 6]. By utilizing higher-order statistics in the
superresolution method, it is reported that a method for
estimating the direction of arrival, with performance ex-
ceeding the traditional method [7], can be developed. With
this as background, we now propose a method of estimation
of the delay time, the direction of arrival, and the signal
strength, by the MMP method utilizing fourth-order cumu-
lants [8, 9].
As the first step, the frequency characteristics of the
multipath propagation environment in mobile radio com-
munication is formulated, and the fourth-order cumulants
used as higher-order statistics are described. Then, the
virtual correlation matrix is generated using the fourth-or-
der cumulants, and a method that estimates the delay time
by the MMP technique is presented. By applying the delay
time estimation process, an estimation method is proposed
for the direction of arrival, the correspondence between the
delay time and the direction of arrival, and the signal
strength. Lastly, the performance of the proposed method
is evaluated by a computer simulation.
2. Multiple Propagation Paths
2.1. Multipath propagation environment in
mobile communications
In a mobile radio communication system, multiple
propagation paths are formed between the mobile station
and the base station as a result of reflection and refraction
by the artificial or natural reflectors around the base and
mobile stations. In other words, the wave received at the
base station is composed of waves arriving with various
delay times and directions of arrival. In addition, not only
reflected waves with a constant deterministic component
within the interval of observation, but also scattered waves
with stochastic fluctuations produced by scattering around
the mobile station, are combined with the above multiple
propagation paths. With the movement of the mobile sta-
tion, rapidly fluctuating fading is produced by the scattered
waves.
Such a multiple propagation path environment is
represented by a model called the GWSSUS (Gaussian
wide-sense stationary uncorrelated scattering) channel [4].
In a time interval corresponding to several tens of wave-
lengths, for which the model is considered valid, there is
little fluctuation in the delay time or the direction of arrival
of the received wave. Since the scattering processes on the
propagation paths are independent, it is assumed that the
fluctuations of the amplitudes and phases of the arriving
waves are mutually uncorrelated [4].
2.2. Formulation of frequency characteristics
of multiple propagation path environment
The frequency characteristics of the multiple propa-
gation path environment are formulated as follows. Con-
sider a straight-line array antenna of L elements with
spacing 'x, as shown in Fig. 1. Let the frequency charac-
teristics of the multiple propagation path environment be
measured at N points at equal intervals 'Z on the frequency
axis, and M points at equal intervals 't on the time axis.
Then, the frequency characteristics H(P, l, i) (=H(w, x, t))
of the multiple propagation path environment at the P-th
frequency and the i-th time are formulated as follows. We
set Z = Z0 + 'Z � P, x = 'x � l, t = 't � i. In the above, Z0 is
the angular frequency of the carrier wave. It is assumed that
the antenna elements have the same characteristics without
directivity, and that there is no interaction between the
antenna elements.
Fig. 1. Linear array antenna.
31
Additive white Gaussian noise n(P, l, i) is assumed
as the observation noise component. Then, the result of
measurement is
The first term in brackets in Eq. (1) is the determinis-
tic reflected component. Let the complex amplitude of the
n-th reflected wave be An and the Doppler angular fre-
quency be Zdn. The second term an
�s��i� is the scattered wave
component with Gaussian stochastic fluctuation. In other
words, the Rice propagation path environment is assumed.
In the above, p is the total number of delayed waves (the
number of propagation paths), an(i) is the complex ampli-
tude of the delayed wave, Wn� is the delay time of the delayed
wave, In is the direction of arrival of the delayed wave, and
c is the speed of light.
Assuming further that the frequency bandwidth of
measurement is sufficiently narrow compared to the carrier
frequency, the following approximation can be applied
where
It is assumed that all mobile stations are sending
signals with the same amplitude at all frequencies that are
measured. For the details of the generation of such signals
and the method of frequency analysis, see Ref. 4.
3. Cumulants
The statistics of third and higher order are called
higher-order statistics (HOS), and their applications to the
signal processing technique have been investigated [5, 6].
Dogan and Mendel [7] presented a method to further im-
prove the resolution by utilizing high-level phase informa-
tion and the Gaussian noise suppression effect of the
fourth-order cumulant, which is one of the higher-order
statistics, in superresolution methods such as MUSIC [2]
and ESPRIT [3]. In the following, the properties of the
statistics called cumulants are described.
3.1. Fourth-order cumulant
For the stochastic variable vector xi with mean 0, the
fourth-order cumulant with 0 as the center is defined as
follows, where E[ ] represents the ensemble average:
Definition (5) can also be applied to the complex stochastic
variable [6].
3.2. Properties of cumulants
Some important properties of the cumulants are now
described. Those properties also apply to the case of the
complex stochastic variables [6].
(1) When a stochastic variable is multiplied by a
(complex) constant, the cumulant itself is multiplied by that
constant. Let Di be a constant and let xi be a stochastic
variable. Then,
(2) The cumulant of the sum of independent stochas-
tic variables is equal to the sum of the respective cumulants
of those variables. When the stochastic variables xi and yi
are mutually independent, we have
(1)
(2)
(3)
(4)
(5)
(6)
(7)
32
(3) The third- or higher-order cumulant suppresses
the Gaussian signal. Let zi be a Gaussian stochastic variable
and let n M 3. Then,
4. Delay Time Estimation by MMP Using
Fourth-Order Cumulant
This paper proposes the following delay time estima-
tion method for multiple delayed waves. The virtual corre-
lation matrix is generated from the result obtained by
processing the frequency characteristics of the multiple
propagation paths using the fourth-order cumulant. The
MMP technique is applied to the virtual correlation matrix
to obtain the estimation [8].
4.1. Processing of frequency characteristics by
fourth-order cumulant
Consider a single antenna element (let l = 0, for
example). The measured frequency characteristic is defined
as follows:
Assume that the above frequency characteristic is
observed at three points on the frequency axis, from the
center frequency P = 0 to the second adjacent point P = 2.
The following fourth-order cumulant is calculated from
three observed values H0 , H1, H2 as shown in Fig. 2 (the
asterisk indicates the complex conjugate):
In the above calculation, property (2) of the cumulant in
Section 3.2 is used, since H and n are statistically inde-
pendent.
We also have
In the above calculation, property (3) is applied to the
scattered wave component and the observed noise compo-
nent. The terms corresponding to the scattered wave com-
ponent and the observed noise component, which have
Gaussian statistical properties, are zero, and there remains
only the term for the deterministic reflected wave compo-
nent. Since zWn
is a constant, property (1) is also used.
Property (2) is used, since it is assumed that there is no
correlation among the multiple delayed waves, as was
discussed in Section 2.1.
When the second-order moment is used, observation
at observation point P = 3 is required in order to determine
the delay term zWn
3 . This is not required when the fourth-order
cumulant is used, as is seen from Eq. (11). There is an effect
that frequency characteristics which are not actually ob-
served are virtually observed.
4.2. Generation of virtual correlation matrix
The virtual correlation matrix is generated in order to
apply the matrix pencil in the MMP method. For two
matrices A and B, the matrix pencil is given by A � zB,
where z is a complex scalar parameter. The special value of
z for which the rank of A � zB is decreased is related to the
generalized eigenvalue, as will be discussed later.
Consider the case where the frequency characteristics
are observed at three points on the frequency axis, as in Fig.
2. Five fourth-order cumulants are calculated so that the
delay times zWn
to the power of 0 to 4 will be included. The
quality Cn defined as follows:
The following relation is clear from the derivation of
Eq. (11):
(8)
(9)
(10)
Fig. 2. Measurement in the frequency domain.
(11)
(12)
(13)
(14)
33
The virtual correlation matrix S is generated by ar-
ranging C0 to C4 as elements of a 5 u 5 matrix. In general,
the virtual correlation matrix of size K is
The virtual correlation matrix S based on the fourth-
order cumulants has a larger size than that of the second-
order correlation matrix. This implies that a larger number
of arriving waves can be estimated.
4.3. Delay time estimation by MMP
The delay times of the arriving waves are estimated
by applying the MMP algorithm [4] to the generated S. The
MMP algorithm is one of the superresolution methods in
which the estimation of the delay time, as well as the
direction of arrival, are reduced to the generalized eigen-
value problem. The peak search process is not required as
in MUSIC. Another feature is that the matrix with just the
rank required for the estimation is used, which helps to
improve the accuracy of estimation. There is not much
difference between TLS-ESPRIT [3] and MMP [4].
The estimation procedure is as follows. By the use of
the fourth-order cumulants, the effect of virtual observation
is realized. Thus, the delay times can be estimated for at
least twice the number of delayed waves, compared to the
traditional method based on the second-order moments.
[Step 1] The virtual correlation matrix is generated
from the observed frequency characteristics. Eigenvalue
decomposition is applied. The number p of delayed waves
is estimated based on the determined rank.
[Step 2] The prediction matrix C(p) is calculated as
follows, where the superscript H indicates the complex
conjugate transpose:
[Step 3] Eigenvalue decomposition is applied to the
prediction matrix C(p). Then, the eigenvalues agree with the
delay terms zWn
�n 1, 2, . . . , p�. [Step 4] The delay times zW
n
of the delayed waves are
calculated as follows from the delay terms Wn (n = 1, 2, . . . ,
p):
m1, M1, M2 in step 2 are derived from the repre-
sentation, which is a compression of the eigenvalue decom-
position of S to rank p. In other words,
The first to the (K � 1)-th columns of matrix M�p�H are
written as M1H, the first column is written as m1, and the
second to the K-th column are written as M2H.
5. Estimation of Direction of Arrival and
Signal Strength by MMP Method Using
Fourth-Order Cumulants
5.1. Estimation of direction of arrival
In the estimation of the direction of arrival, the direc-
tion is observed as the time difference of the arrivals at the
antenna elements. The virtual correlation matrix is gener-
ated for the direction of arrival zIn
of the delayed wave, and
the direction of arrival is estimated by applying the MMP
algorithm. The procedure is the same as in the estimation
of the delay time discussed in Section 4.3. Based on the
estimated direction of arrival terms zIn
�n 1, 2, . . . , p�,the directions of arrival of the delayed waves I
n (n = 1, 2,
. . . , p) are calculated as follows:
Using the fourth-order cumulants, the effect of virtual
observation of the frequency characteristics is realized for
antenna elements which do not actually exist [7]. Compared
to the case where the second-order moment is used, the
directions of arrival can be estimated for at least twice the
number of delayed waves [9].
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
34
5.2. Estimation of correspondence between
delay time and direction of arrival
The frequency characteristics measured at multiple
frequencies in the straight-line array antenna can be repre-
sented as follows:
Assume that the above frequency characteristics are ob-
served at the observation points on the oblique line in Fig.
3. Then, the correspondence term zWn
zIn
can be estimated by
MMP, as in the estimation of the delay times.
In order to establish the correspondence between the
delay time Wn and the direction of arrival I
n for multiple
delayed waves, the product zWizIj
of the estimated delay
term zWi �i 1, 2, . . . , p� and the estimated direction of
arrival term zIj �j 1, 2, . . . , p� is calculated. When the
result agrees with one of the estimated correspondence
terms zWkzI
k (k = 1, 2, . . . , p), it is estimated that the delay
time Wi corresponds to the direction of arrival Ij [9].
5.3. Estimation of signal strength
The matrix element Cn of the virtual correlation ma-
trix generated for the estimation of the delay time is related
to the complex amplitude An of the reflected wave compo-
nent of the delayed wave as follows. A similar relation also
exists in the case of the virtual correlation matrix for the
estimation of the direction of arrival:
By calculating the square root of |An|
4 estimated by
Eq. (24), the signal strength |An|
2 of the reflected wave
component can be estimated.
6. Reduction of Error in Virtual
Correlation Matrix
The cumulant was originally defined as the ensemble
average for the ensemble of the stochastic processes. In the
proposed method, however, the average is replaced by the
time average for a certain sample function over a finite time.
Because of this situation, when the fourth-order cu-
mulant is calculated from a finite sampled value sequence,
the result may deviate greatly from the true value. Due to
this deviation, the virtual correlation matrix composed of
fourth-order cumulants estimated from a finite number of
samples includes an error. The technique used to reduce the
computation error in the virtual correlation matrix is shown
in the following.
6.1. Reduction of error of virtual correlation
matrix
When the observation is made as in Fig. 2, the value
of a single fourth-order cumulant is used as the matrix
element C2, as in Eq. (15). When three data for the fre-
quency characteristics are observed, however, several val-
ues for the fourth-order cumulant with the same phase
information zWn
2 can be calculated. From such a viewpoint,
the arithmetic mean of several fourth-order cumulants with
the same phase information is used as the matrix element
of the virtual correlation matrix:
(23)
Fig. 3. Observation points.
(24)
35
6.2. Reduction of error of virtual correlation
matrix 2
In the calculation of the fourth-order cumulants as the
matrix elements of the virtual correlation matrix, the fre-
quency characteristics are observed for the same time se-
quence, as in Eq. (15). In the calculation of the reflected
wave component, however, the phase information is main-
tained the same, even if each of the two frequency charac-
teristics is shifted by k [10]. Consequently, the time series
are shifted m times, and the arithmetic mean of the results
is used as the matrix element of the virtual correlation
matrix:
7. Performance Evaluation
7.1. Parameters of computer simulation
The performance of the proposed method was evalu-
ated by computer simulation. Table 1 shows the parameters
of the multiple delayed waves (signal waves) used as the
object of estimation. The exponential type [11] is used as
the delay profile. The Rice coefficient is the power ratio
between the deterministic reflected signal component and
the scattered signal component with stochastic fluctuation.
For the observation noise, the signal power is summed up
for all arriving waves, and six values are used, where the
SNR (signal power-to-observed noise power ratio) is varied
from 5.0 dB to 30.0 dB in 5.0-dB steps.
It is assumed that the mobile station (transmitting
station) is moving at a speed of 40.0 km/h. The Doppler
frequency shift is assumed for each transmitting direction
from the mobile station. It is assumed that there is no direct
wave to the base station, and no particular direction of
motion with respect to the base station is considered.
In the estimation of the signal waves of Table 1, it is
assumed that the frequency characteristics are observed at
three points on the frequency axis by a straight-line array
antenna with three antenna elements. Figure 3 shows the
placement of the observation points, and Table 2 shows the
sampling parameters. A 7 u 7 virtual correlation matrix is
generated from the data. One thousand sampling data are
used to calculate the fourth-order cumulants. Forty-eight
time-series shifts are applied in the error reduction dis-
cussed in Section 6.2. The total number of sampled data is
1048 u 7 = 7336.
7.2. Result of estimation
In practice, the number p of delayed waves must be
estimated. In this simulation, however, this process is omit-
ted. The information that p = 4 is given beforehand, and the
estimation is performed.
Figures 4 to 6 show the estimates of the delay time,
the direction of arrival, and the signal strength, respectively.
One hundred trials are made for each SNR value. In Fig. 6,
the signal strength is normalized to the power of the firstTable 1. Arriving waves
1st
wave
2nd
wave
3rd
wave
4th
wave
Center frequency
[GHz]
1.9
Delay time [Ps] 0.6 1.0 1.9 2.6
Delay time differ-
ence [Ps]
0.0 0.4 0.9 0.7
Direction of arrival
[degree]
�10.0 8.0 25.0 �30.0
Signal strength [dB] 0.0 �3.0 �3.7 �4.0
Rice coefficient 2.0
Moving velocity
[km/h]
40.0
(25)
(26)
Table 2. Sampling items
Observation point interval on
frequency axis
[kHz] 300
Observation bandwidth on
frequency axis
[kHz] 900
Element interval of array antenna [cm] 7.9
Size of array antenna [cm] 23.7
Number of sampling points on
time axis
[samples] 1048
Sampling interval on time axis [ms] 1.0
36
wave. The standard deviation of the third wave is omitted
since it overlaps with those of the second and fourth waves.
As to the estimation of the correspondence between
the delay time and the direction of arrival, the success
probability exceeds 95% for SNRs above 10 dB. The esti-
mation is treated as a success when the estimated direction
of arrival roughly agrees with the true direction of arrival.
When the SNR is 5.0 dB (the power of the observation noise
is 0.4 dB larger than the power of the first reflected compo-
nent), there is a failure rate of 8%. It is seen from Fig. 4 that
the first and second waves can be estimated with small
variance and the results are close to the true values. The
third and fourth waves, with small power, are estimated
with large variance. The result, however, is unbiased. In
other words, the average of a large number of estimated
results (100 in this case) agrees with the true value.
The tendency is similar in the results of estimation of
the direction of arrival shown in Fig. 5. The direction of
arrival can be estimated for four waves, which is more than
the number of antenna elements, i.e., three. When the
number of observation points on the frequency axis is three,
and the number of antenna elements in the straight-line
array antenna is three in the estimation of the delay times
and the directions of arrival, a maximum of two waves can
be handled by the traditional MMP algorithm using the
second-order moment. In other words, it is shown that the
proposed method can estimate twice as many delayed
waves.
In the estimation of the signal strength shown in Fig.
6, on the other hand, the variance of the estimated value is
large, but it is seen that the power of the reflected compo-
nents of the delayed waves can be roughly estimated. It
should be noted that when the error reduction techniques
described in sections 6.1 and 6.2 are not used, estimation of
both the delay time and the direction of arrival is completely
impossible.
8. Conclusions
The following method is proposed. The frequency
characteristics of the multiple propagation path environ-
ment are observed by a straight-line array antenna. The
observed data are processed by the method of fourth-order
cumulants and the virtual correlation matrix is generated.
By applying the MMP technique to the virtual correlation
matrix, the delay time, the direction of arrival, the corre-
spondence between the delay time and the direction of
arrival, and the signal strength are estimated.
The proposed method uses the fourth-order cumu-
lants. As a consequence, when the number of observation
Fig. 4. Delay time estimation.
Fig. 5. DOA estimation.
Fig. 6. Signal strength estimation.
37
points on the frequency axis and the number of antenna
elements of the straight-line array antenna are the same, the
method can estimate at least twice as many delayed waves
as in the case of the second-order moment. Next, a tech-
nique to reduce the error of the virtual correlation matrix is
proposed. The performance is evaluated by a computer
simulation, and the usefulness of the proposed method is
demonstrated. In this paper, MMP is used as the superreso-
lution technique, and the virtual correlation matrix based
on the cumulants is applied. The virtual correlation matrix
can in principle be introduced into other well-known super-
resolution methods such as MUSIC and ESPRIT. This point
will be investigated in the future, together with the problem
of estimating the number of delayed waves from the rank
of the virtual correlation matrix.
Acknowledgments. Part of this study was sup-
ported by the Elect. Comm. Foundation, SCAT Research
Grant and Science Grant, Ministry of Education. The
authors acknowledge the suggestions and advice made in
this study by Professor K. Yamada, Dept. Func. Mat. Eng.,
as well as by members of the Information Processing Cen-
ter, Saitama University.
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AUTHORS
Yuichi Shimamura (member). Education: Saitama University, Department of Electronic Engineering, B.E., 1995;
doctoral program. Industry positions: NTT. Research interests: digital signal processing.
38
AUTHORS (continued) (from left to right)
Kiyomichi Araki (member). Education: Saitama University, Department of Electrical Engineering, B.E., 1971; Tokyo
Institute of Technology, M.E., 1973, D. Eng., 1978. Academic appointments: Tokyo Institute of Technology, research associate,
1978; Saitama University, associate professor, 1985; University of Texas, visiting researcher, 1979�80; Saitama University,
associate professor, 1985; University of Illinois visiting researcher, 1993�94; Tokyo Institute of Technology, Faculty of
Engineering, professor, 1995. Research interests: information security, code theory, information theory, circuit theory, micro-
wave circuits. Memberships: IEEE, Information Processing Society of Japan.
Jun-ichi Takada (member). Education: Tokyo Institute of Technology, Department of Electrical and Electronic Engi-
neering, B.E., 1987, D. Eng., 1992. Academic appointments: Chiba University, Faculty of Engineering, research associate;
Tokyo Institute of Technology, International Cooperative Center for Science and Technology, associate professor. Research
interests: planar antennas, small antennas, mobile propagation; academic cooperation with Southeast Asia. Memberships: IEEE,
ACES, SIAM. Recognitions: IEICEJ Paper Award, 1992; IEICEJ Young Engineer Award, 1993.
39